I am not licensed in IL. This is a general discussion of the issues only and is not intended, nor should it be construed, as legal advice.
Since you did not attend court, you were essentially deemed to have admitted the charge against you, and since it was a traffic-related issue, the suspensions are the penalty. The $300-$400 fine may have been the fine that correlates with not producing proof of insurance, but that is an issue of IL law on which I cannot comment.
You may wsh to contact the Clerk of the Court (where the hearing had been set) and ask the procedure for resetting the hearing; there may be a fee to reset it. Follow the process, pay the fee, and get the hearing reset; attend, and bring with you proof of insurance at the time of the ticket and now. If you can produce proof of insurance at all times material, that may be the end of it.